Efficient algorithms for computing a minimal homology basis
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Efficient computation of shortest cycles which form a homology basis under Z_2-additions in a given simplicial complex K has been researched actively in recent years. When the complex K is a weighted graph with n vertices and m edges, the problem of computing a shortest (homology) cycle basis is known to be solvable in O(m^2n/ n+ n^2m)-time. Several works borradaile2017minimum, greedy have addressed the case when the complex K is a 2-manifold. The complexity of these algorithms depends on the rank g of the one-dimensional homology group of K. This rank g has a lower bound of Θ(n), where n denotes the number of simplices in K, giving an O(n^4) worst-case time complexity for the algorithms in borradaile2017minimum,greedy. This worst-case complexity is improved in annotation to O(n^ω + n^2g^ω-1) for general simplicial complexes where ω< 2.3728639 le2014powers is the matrix multiplication exponent. Taking g=Θ(n), this provides an O(n^ω+1) worst-case algorithm. In this paper, we improve this time complexity. Combining the divide and conquer technique from DivideConquer with the use of annotations from annotation, we present an algorithm that runs in O(n^ω+n^2g) time giving the first O(n^3) worst-case algorithm for general complexes. If instead of minimal basis, we settle for an approximate basis, we can improve the running time even further. We show that a 2-approximate minimal homology basis can be computed in O(n^ω√(n n)) expected time. We also study more general measures for defining the minimal basis and identify reasonable conditions on these measures that allow computing a minimal basis efficiently.
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